Methods for assigning a price to an asset that is a derivative of a non-marketed variable

ABSTRACT

A computer-implemented method is provided for valuing and hedging payoffs that are determined by an underlying non-marketed variable that moves randomly [ 200 ]. The value assigned is that which is obtained by projecting the instantaneous return of the future payoff onto the span of marketed assets. An explicit method is provided for determining this value by determining a suitable market representative [ 210 ]. In a continuous-time embodiment, the methodology is based on an extended Black-Scholes equation [ 220 ] that accounts for the correlation between the underlying non-tradable asset and marketed assets. Once this extended equation is solved [ 210 ], the value of the payoff, the optimal hedging strategy [ 240 ], and the residual risk of the optimal hedge can be determined [ 250 ]. In alternate embodiments, the same value is determined as the discounted expected value of the payoff, using risk-neutral probabilities for the non-marketed variable. These risk-neutral probabilities are again determined by the relation of the underlying variable to the payoff of a most-correlated marketed asset. The risk-neutral version of the method applies in both continuous-time and discrete-time frameworks, providing asset valuation, optimal hedging, and evaluation of the minimum residual risk after hedging.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority from U.S. Provisional PatentApplication No. 60/395,715 filed Jul. 12, 2002, which is incorporatedherein by reference.

FIELD OF THE INVENTION

[0002] This invention relates to computer-implemented methods within thefield of finance. More specifically, the invention relates to methodsdirected at valuing complex uncertain cash flow streams and determiningthe best way to hedge exposure to such streams.

BACKGROUND

[0003] Frequently it is desirable to assign a price to an asset that isnot currently traded, or to independently verify a price of an assetthat is traded. There are two standard methods for assigning such prices(both of which earned Nobel prizes). The first is the capital assetpricing model (CAPM) [1] which is a method for assigning a price to anasset that has a duration of one period of time (such as one year) andis not traded at intermediate times. The second is the Black-Scholesmethod [2] for pricing financial derivatives, which assigns a price to aderivative (such as an option) on a underlying asset that is traded.

[0004] A derivative security is a security whose payoff is determined bythe outcome of another underlying security. For example, a stock optionon a traded stock is a derivative, since the final payoff of the optionis completely determined by the value of the stock at the terminal time.The Black-Scholes equation is the standard method for determining thevalue of derivatives. It is based on the fact that, in continuous time,it is possible to replicate the payoff of a derivative by a portfolioconsisting of the underlying security and a risk free asset (such as aU.S. Treasury bill) with a fixed risk free interest rate r. Thefractions of the portfolio devoted to each of its components is adjustedcontinuously so that the portfolio's response to changes in theunderlying security will perfectly mirror the response of thederivative. This adjustment process is termed a replication strategy. Itis argued that the value of the derivative is equal to the cost ofreplication strategy. The cost V is determined by the Black-Scholesequation, which gives the cost (or value) V(x,t) for values of x≧0 and0≦t≦T where T is the terminal time of the derivative. Specifically, theequation is

rV(x, t)=V _(t)(x, t)+V _(x)(x, t)rx+½V _(xx)(x, t)x ²σ².   (1)

[0005] Here x denotes the value of the underlying security, r is theannual risk free interest rate, and σ² is the annual volatility of theunderlying security. The notations V_(t), V_(x), V_(xx) denote,respectively, the first partial derivative with respect to t, and to x,and the second partial derivative with respect to x. The equation issolved with the boundary condition V(x, T)=F(x(T)), where F denotes thepayoff at time T of the derivative. For example, if the derivative is acall option with strike price K, then F(x(T))=max (x(T)−K, 0).

[0006] A more general situation is where a payoff depends on a variablex_(e) but this variable is not traded. For example, an option dependingon a firm's revenue is of this form, because the revenue (which is x_(e)in this case) is not traded. Hence the payoff depends on a non-tradedunderlying variable. In these situations it is impossible to form areplicating strategy using the conventional Black-Scholes equationbecause it is impossible to trade the underlying variable. Theassumption underlying the conventional Black-Scholes equation breaksdown.

[0007] Such problems have been studied by other researchers. The idea ofusing a market hedging strategy to minimize the expected squared errorbetween the final value of the hedge and the actual payoff was proposedby Föllmer and Sondermann [3], who showed that it was possible inprinciple. Because minimizing the expected squared error is equivalentto orthogonal projection of the payoff onto the space of marketedpayoffs (under a standard definition of projection), the method is oftenreferred to as projection pricing. Their analysis, however, is purelyabstract and does not exhibit any practical method for explicitlyfinding the hedge by solving a partial differential equation or adiscrete version of it.

[0008] Merton (in his Nobel Prize acceptance speech [6]) emphasized theimportance of the problem. He proposed a procedure based on the originalBlack-Scholes equation, but it is essentially an ad hoc method that doesnot coincide with projection. He and Pearson [7] studied a generalframework of incomplete markets and formulated prices based on aconsumer maximization problem. Again their method is abstract and doesnot provide a direct formula for the value that could be used inpractice.

[0009] Some practitioners have proposed various ad hoc translations ofthe standard Black-Scholes equation to specific situations. For example,it is common practice to artificially increase the volatility of theunderlying variable in an attempt to recognize that the variable is notreally a traded asset. These methods are not based on optimality, do notfundamentally revise the original Black-Scholes equation, nor have anyother real theoretical basis.

[0010] An approach that directly addresses the problem presents asolution in terms of a market price of risk, which is applicable to allderivatives of a non-marketed variable. The market price is difficult tomeasure, but it has sometimes been estimated [8]. However, even if themarket price of risk is known, it does not lead to a hedging strategy.

[0011] Overall, there has not been an effective and practical methodproposed that prices, optimally hedges, and computes the residual risk(after hedging) of derivatives of non-marketed variables.

SUMMARY OF THE INVENTION

[0012] Embodiments of the invention provide computer-implementedmethods, based on projection, for determining the value of a derivativeof a non-traded variable. In addition, some embodiments provide a methodfor determining the optimal hedging strategy and its residual risk.

[0013] Several embodiments of the invention are centered on a new,extended Black-Scholes equation of the form $\begin{matrix}\begin{matrix}{{{rV}\left( {x_{e},t} \right)} = {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}{x_{e}\left\lbrack {\mu_{e} - {\beta_{em}\left( {\mu_{m} - r} \right)}} \right\rbrack}} +}} \\{{{\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}x_{e}^{2}\sigma_{e}^{2}},}}\end{matrix} & (2)\end{matrix}$

[0014] where

β_(em)=ρ_(em)σ_(e)/σ_(m).

[0015] The underlying variable x_(e) is not marketed, (or equivalently,not traded). By this we mean, throughout, that it is not marketed as asecurity that can be held without cost and pays no dividend while held.Thus, although oil can be bought and sold, it is not traded in thissense because there are storage costs. However, a futures contract onoil is traded as a security. In some cases, a variable that is nottraded in our sense can be converted to a traded asset; (for example, adividend-paying stock can be so converted by subtracting the presentvalue of its dividend). Such cases do not require the methods of thisinvention (although the methods of this invention may lead more quicklyto the correct result). If our methods are applied to a case where theunderlying variable is in fact traded, the extended equation will reduceto the standard Black-Scholes equation.

[0016] The extended equation (2) uses, in addition to the underlyingvariable x_(e), an asset x_(m) that serves as a market representative.This representative is itself a marketed variable, or a linearcombination of marketed assets, in the sense described above. Thisvariable enters the equation indirectly through specification of certainparameters of the equation. The constants μ_(e) and μ_(m) are the driftrates of x_(e) and x_(m), respectively. ρ_(em) is the correlationcoefficient between x_(e) and x_(m). σ_(e) ² and σ_(m) ² are the annualvariances of x_(e) and x_(m), respectively.

[0017] The replacement of r by [μ_(e)−β_(em)(μ_(m)−r)] as a coefficientof V_(x) _(e) (x_(e), t)x_(e) in the partial differential equationdistinguishes this extended equation from the conventional Black-Scholesequation. This coefficient accounts for the fact that perfectreplication is not possible in all cases.

[0018] The market representative x_(m) can be chosen in several ways,each of which leads to an identical result. In some embodiments x_(m) ischosen to be the Markowitz (or market) portfolio of risky assets.According to other embodiments, x_(m) is chosen to be a portfolio ofmarketed securities most correlated (related) to the underlying variablex_(e). This method is simpler than choosing x_(m) to be the Markowitz(market) portfolio and furthermore leads to an optimal mean-squarehedging strategy.

[0019] Embodiments of the invention include an implementation based on asolution to a continuous-time version of the extended Black-Scholesequation. Thus, according to an embodiment, a method is provided forpricing a financial derivative of a non-marketed variable represented byx_(e). The method includes determining a market representative x_(m)that is useful in determining the value of the financial derivative.Information associated with the non-marketed variable and the marketrepresentative is retrieved. A solution to an equation involving avariable V(x_(e), t) representing a price of the financial derivative isthen calculated. The equation, which in this embodiment is preferablythe extended Black-Scholes equation, includes a coefficient involvingthe information associated with the market representative x_(m) and thenon-marketed variable x_(e). An output including a calculated price ofthe financial derivative is then generated. Another embodiment includesa solution using formulas based on risk-neutral processes.

[0020] The price function can also be found in a discrete-timeimplementation. The underlying process can be discretized directly, or afinite-state model can be used. For example, in an embodiment of theinvention, a method is provided for pricing a financial derivative of anon-marketed variable represented by x_(e). The method includesretrieving information associated with a suitable market representativex_(m). A solution is then found to a system of equations involving avariable V(x_(e), k) representing a price of the financial derivative atdiscrete times indexed by k. The equations include a coefficientinvolving the information associated with the variable x_(m) associatedwith the market representative. An output including a calculated priceof the financial derivative is then generated.

[0021] In another embodiment, a computer-implemented method is providedfor pricing a financial derivative of a non-marketed variablerepresented by a finite-state variable B. The method includesdetermining a market representative, represented by a variable A.Risk-neutral probabilities are then calculated using a binomial latticemodel associated with the non-marketed variable and the marketrepresentative. Values of a variable V on the lattice corresponding tothe variable B are also calculated. The calculated values of V representa calculated price of the financial derivative.

[0022] Embodiments of the invention may also involve computing anoptimal hedge and other variables of interest. The above embodiments maybe easily implemented in various different ways on any type of computer,and may be realized as instructions stored on a computer-readablemedium.

BRIEF DESCRIPTION OF THE DRAWINGS

[0023]FIG. 1 presents a schematic related to a continuous-timeimplementation of the invention. It shows the key variables and how theycombine to produce the value of the derivative of the underlyingvariable x_(e).

[0024]FIG. 2 is a flow diagram showing the steps of the method in thecontinuous-time case.

[0025]FIG. 3 is a schematic diagram illustrating the projection of V+dVonto the space M.

[0026]FIG. 4 shows a single step of a binomial model of variable A withprobabilities ρ_(A) and 1−ρ_(A).

[0027]FIG. 5 shows a single step of a binomial model of a variable Bwhen converted to risk-neutral form with risk-neutral probabilities ofq_(B) and 1−q_(B).

[0028]FIG. 6 presents a schematic of the method for the discrete-timecase. It shows the key variables and how they combine to produce thevalue of the derivative of the underling variable B.

[0029]FIG. 7 is a flow diagram showing the steps of the method in thediscrete-time case.

DETAILED DESCRIPTION: OVERVIEW

[0030] Extended Black-Scholes Equation

[0031] Several embodiments of the invention are based on solving anextended Black-Scholes equation. This novel equation differs from thestandard Black-Scholes equation in that it involves a new coefficient inone of its terms. This coefficient allows the applications of theequation to be extended to include situations that the standardBlack-Scholes equation fails to cover.

[0032] A schematic depiction illustrating aspects of an embodiment ofthe invention is presented in FIG. 1. The figure shows an underlyingvariable x_(e), which is specified at 100. The variable x_(e) defines anew asset payoff through a function F. Also shown in the figure is amarket representative x_(m), determined at 110. The marketrepresentative x_(m) is extracted from marketed securities, which aredetermined at 120. If this market representative is most correlated withx_(e), it will depend on x_(e); and this is indicated in the figure bythe dashed line from x_(e) to the market representation choice process,110. The properties of this representative and x_(e) define theparticular coefficients of a partial differential equation, 130. Thefunction F defines the terminal boundary condition at 140. The solutionof the equation then yields the value of the new asset at 150.

[0033] Although the new method may at first appear arbitrary since it isbased on projection, it has a compelling justification. TheBlack-Scholes approach uses replication to render the new payoffredundant, in the sense that the payoff is already embedded in thesecurities market. In a similar way, the new approach approximates thepayoff and renders the payoff irrelevant in the sense that norisk-averse investor will want to own the new payoff because it isinferior to things already embedded in the securities market.

[0034] Optimal Hedge

[0035] Once the value function V(x_(e), t) is found, it is possible tofind the portfolio strategy that best approximates the new asset, andcan be used as a hedging strategy.

[0036] When the hedging portfolio has value H(x_(e), t), an amount(H(x_(e), t)−φ) is invested in the risk free asset and an amount φ isinvested in the most-correlated marketed asset where

φ(x_(e), t)=V_(x)(x_(e), t)x_(e),β_(ec).   (3)

[0037] The minimum variance of the error V(x_(e), T)−H(x_(e), T) can befound by solving another partial differential equation, namely,$\begin{matrix}{{{{S_{t}\left( {x_{e},t} \right)} + {{S_{x_{e}}\left( {x_{e},t} \right)}\mu_{e}x_{e}} + {\frac{1}{2}{S_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}x_{e}^{2}} + {{^{2{r{({T - t})}}}\left\lbrack {{V_{x_{e}}\left( {x_{e},t} \right)}\sigma_{e}x_{e}} \right\rbrack}^{2}\left( {1 - \rho_{ec}^{2}} \right)}} = 0},} & (4)\end{matrix}$

[0038] with boundary condition S(x_(e), T)=0. The value S(x_(e), 0) isthe variance of the replication error at T, as seen at t=0.

[0039] Risk-Neutral Form

[0040] An embodiment of this invention is an implementation as apractical risk-neutral valuation method. In this version the drift ofthe underlying variable x_(e) is changed from μ_(e) to[μ_(e)−β_(em)(μ_(m) −r)].

[0041] The general formula for value is then

V(x _(e) , t)=e ^(−r(T−t)) Ê[F(x _(e)(T))],   (5)

[0042] where Ê denotes expectation with respect to the risk-neutralmodel. Hence, in this version, the value is simply the discountedexpected value of the final payoff, with expectation taken with respectto the risk-neutral model.

[0043] The advantage of this form of the valuation is that value can beestimated by simulation. Basically, thousands of simulation runs of theunderlying variable are made and the average discounted value is used asthe expected discounted value. Such simulation methods are standardpractice. A new feature of this embodiment is specification of therisk-neutral equation.

[0044] Discrete-Time Version

[0045] The risk-neutral form can be adapted to a discrete-time frameworkwhere the dynamics of random variables are represented by lattices. Inthe basic model, the underlying variable B is assumed at each step toeither go up (with probability p_(B)) or down (with probability 1−p_(B))A payoff that is function of this underlying variable can be evaluatedby using the risk-neutral probabilities, denoted q_(B) and 1−q_(B).These, in turn, are found by using a market representative as in thecontinuous-time case. If this market representative is denoted A withprice (value) ν_(A), the risk-neutral probability q_(B) is$\begin{matrix}{{q_{B} = {p_{B} - {{{{cov}\left\lbrack {A,{1\left( U_{B} \right)}} \right\rbrack}\left\lbrack {\overset{\_}{A} - {\upsilon_{A}R}} \right\rbrack}/\sigma_{A}^{2}}}}\quad} & {(6)} \\{= {p_{B} - {\beta_{{1{(U_{B})}},A}\left\lbrack {\overset{\_}{A} - {\upsilon_{A}R}} \right\rbrack}}} & {(7)}\end{matrix}$

[0046] where 1(U_(B)) is a payoff of 1 if B is up, and 0 if B is down.The expression “cov” stands for covariance, and μ_(1(U) _(B) _(),A) isdefined as cov(1(U_(B)), A)/variance(A).

[0047] A derivative of B is a payoff with a value of G_(u) if up occursand payoff G_(d) if down occurs. The current price of such a (future)payoff is $\begin{matrix}{V_{B} = {{\frac{1}{R}\left\lbrack {{q_{B}G_{u}} + {\left( {1 - q_{B}} \right)G_{d}}} \right\rbrack}.}} & (8)\end{matrix}$

[0048] This is the basic method for the discrete-time case. The uniquefeature is the formula (6) for the risk-neutral probability.

[0049] Recursive Solution and Hedging

[0050] If the payoff occurs at the end of some finite number of steps(at the end of a lattice structure for B), the value function can befound recursively. The recursive solution is $\begin{matrix}{{{V_{k - 1}\left( s_{k - 1} \right)} = {\frac{1}{R}\left\lbrack {{q_{B}{V_{k}\left( s_{k - 1}^{u} \right)}} + {\left( {1 - q_{B}} \right){V_{k}\left( s_{k - 1}^{d} \right)}}} \right\rbrack}},} & (9)\end{matrix}$

[0051] where s_(k−1) ^(u) denotes the upper successor state to s_(k−1)and s_(k−1) ^(d) denotes the lower successor state to s_(k−1). Theprocess is started with the terminal boundary condition specifying thepayoff of the derivative G.

[0052] An optimal hedge H can be found. If at the beginning of timeperiod k the hedge portfolio's value is H_(k−1), then H_(k−1)−γV_(k−1)is invested in the risk free asset and γV_(k−1) is invested in themost-correlated asset A, where

γ=cov(V _(k) /V _(k−1) , A)/σ_(A) ²   (10)

[0053] The variance associated with optimal hedging can also be found bya recursion, as discussed later.

[0054] A Complete Methodology

[0055] Embodiments of the invention provide complete methodologies fortreating payoffs that are derivative of non-traded variables. It has astrong justification and a straightforward representation. For manyproblems, the new method can be put into practice by augmenting standardmethods of financial computation. The requisite equations or latticestructures are similar to those derived from the Black-Scholesmethodology and hence the computational methods are similar. These aremethods for solving partial differential equations, methods forevaluating lattices, and simulation methods. A significant additionalstep of the new method is determination of the market representativex_(m), and estimation of the constants μ_(e), μ_(m) (the drifts), thevariance of the market representative and the value of β_(em).

[0056] Applications

[0057] Embodiments of this invention have applications in numerousfinancial areas. For example, the methods may serve four functions. Thefirst is pricing, as in determining the fair value of an option on anon-marketed variable. The second is design, as in the design of optimalcontracts, project plans, business arrangements, and various agreements.The third is hedging, to minimize the risk associated with a non-tradedasset by use of offsetting market participation. The fourth function isthat of risk assessment, determining the residual risk after optimalhedging. In all cases, a financial payoff is tied to the performance ofa variable that moves randomly but which is not a marketed financialsecurity. The variable that defines the payoff is termed the underlyingvariable.

[0058] The key idea is that when the underlying variable of an asset isnot marketed, and hence the standard Black-Scholes equation does notapply, a market representative can be used instead.

[0059] One large area of application is to the pricing and hedging of“off-exchange” derivatives. For example, it may be desirable to hedgegrapefruit production. Here the underlying variable is grapefruit pricebut there may be no financial instrument (such as a grapefruit futurescontract) that is directly related to this underlying variable. Hencegrapefruit production is not a derivative security and the Black-Scholestheory is not applicable. Nevertheless, a combination of orange juicefutures contracts may be most-correlated with grapefruit prices and thuscan serve as the market representative. This representative can be used(according to the valuation equation) to find value and also serve asthe basis for an optimal (but imperfect) hedging strategy. Similarsituations arise with off-exchange trading of energy contracts,agricultural products, metals, foreign exchange, and many others.

[0060] One important example is that of real estate projects. Typically,there is no underlying security for a real estate project, and thus theBlack-Scholes equation does not apply. However, the payoff of such aproject is likely correlated with interest rates, real estate investmenttrusts, and regional economic variables. A combination of these canserve as the appropriate market representative and hedging strategy.

[0061] Another general area of application is the determination oflong-term futures prices. Typically, futures contracts extend severalmonths, or at best a few years. Yet, many large projects span severalyears and correct pricing and hedging is important. The methods of thisinvention can be used to obtain such prices and hedges.

[0062] Bonds are often issued that have unique risk, not completelycorrelated to marketed securities. For example, a corporate bond's pricemay be only loosely correlated to interest rates and to the stock priceof the associated firm and to the stock prices of firms in the same orrelated industry. Hence the bonds are not perfect derivatives, but themethods of this invention can be used to evaluate them. Large physicalprojects, such as oil rigs, dams, various other infrastructure projectsand private or public works have cash flows that are to some extentassociated with risks that are not derivatives of market securities.Projects in foreign countries have associated country risk, which canonly partially be hedged with bonds and insurance.

[0063] Contracting is another large area of application. For example, anelectronics firm may plan to purchase a great deal of DRAM during theyear. The firm can design contracts with options and price caps thatprovide reduced risk at favorable prices. The design of such a contractcan be deduced from the market representative and the methods of thisdocument. Contracts for movie and publishing rights can also be designedthis way.

[0064] The purchase of intellectual property rights through licenses isa fertile area of application. For example, a pharmaceutical firm maywish to license a biochemistry patent from a university, and both thedesign and the pricing of such a license could be carried out with themethods. In this case, the market representative may be a combination ofthe security prices of other pharmaceutical companies, HMOs, and otherinsurance companies.

[0065] Insurance and various guarantees generally embody a great deal ofrisk. Such risk can be priced and hedged by the insurance company orguarantor by these methods. The risk of fire in an area may, forexample, be correlated with weather derivatives and energy prices.

[0066] Mortgages contain interest rate risk, but also prepayment risk.The prepayment risk may be correlated (imperfectly) to stock marketindices, real estate prices, as well as interest rates.

[0067] Real options is a general term which includes options related tomany of the examples listed above, but especially to business projects.The methods of this document can improve both the evaluation and designof such projects by using a theoretically justified valuation formularather than an ad hoc use of the Black-Scholes equation.

[0068] Another important area of application is that of overall riskassessment. The residual risk of each individual project or derivativecan be determined by the variance given as the solution to a partialdifferential equation that is used in embodiments of the invention. Therisk associated with a collection of such projects can be found bysolving a similar equation that depends on the value function for eachof the projects separately.

DETAILED DESCRIPTION:

[0069] Continuous-Time Case

[0070] This section presents example application areas and gives adetailed description in both the continuous-time and discrete-timeframeworks.

[0071] Outline of Steps: Continuous-Time Framework

[0072] An outline of the steps used in one embodiment of the inventionare indicated in FIG. 2.

[0073] 1. Set Up. In this step, 200, we identify an asset of interestwhich gives a future payoff (or payoffs) according to the value of somevariable that moves randomly with time.

[0074] This variable is termed the underlying variable. If this variableis itself traded (and is governed by geometric Brownian motion (GBM)),then the asset of interest is a pure derivative and the standardBlack-Scholes equation can be used. If the underlying variable is nottraded, then the extended Black-Scholes equation is used.

[0075] For example, an asset might be the future harvest of a grapefruitfarm. The underlying variable determining payoff is grapefruit pricewhich varies randomly from the current time until harvest. Thisunderlying variable cannot be traded. It is true, of course, thatgrapefruit can be bought and sold at any time, but grapefruit cannot beheld as a security since there are storage costs and grapefruit isperishable. The grapefruit price is denoted x_(e) and is assumed tofollow a GBM process of the form

dx _(e)=μ_(e) x _(e) dt+σ _(e) x _(e) dz _(e)   (11)

[0076] where z_(e) is a standardized Wiener process. The process startsat time 0. The asset to be priced has a payoff at time T of F(x_(e), T).

[0077] There are n (perhaps thousands of) securities that can be traded.Each of these follows a similar processes, namely,

dx _(i)=μ_(i) x _(i) dt+σ _(i) x _(i) dz _(i)   (12)

[0078] where x_(i) is the value of the i-th asset, for 1=1, 2, . . . ,n. The standardized Wiener processes z_(i) (of zero mean and unitvariance) are correlated, with cov(dz_(i), dz_(j))=ρ_(ij)dt. Among thesen assets there is an asset that is risk free at each instant. By formingan appropriate combination of the marketed assets, it may be assumedwithout loss of generality that the risk free asset is x_(n). Then

dx_(n)=rx_(n)dt.

[0079] In the standard Black-Scholes method, only the underlyingsecurity, the risk free rate r, and the payoff function is specified. Noconsideration is given to other securities, because the underlyingsecurity is all that is needed. In embodiments of this invention, othersecurities are considered in order to select a market representative, asexplained in the next step.

[0080] 2. Determination of a Market Representative. The extendedBlack-Scholes equation uses a market representative x_(m), which is acombination of marketed assets. In this step, 210, this marketrepresentative is determined. Generally, this market representative canbe chosen in one of several ways.

[0081] (a) A most-correlated asset is a marketed asset x_(c) (or acombination of marketed assets) whose instantaneous return is mostcorrelated (or close to most correlated) with dx_(e). In theory, it isobtained by solving the equations $\begin{matrix}{{\sum\limits_{j = 1}^{n - 1}{\sigma_{ij}\alpha_{j}}} = \sigma_{ie}} & (13)\end{matrix}$

[0082] for each i=1, 2, . . . n−1. The resulting α_(i)'s are thennormalized to sum to 1. The most-correlated asset is the asset x_(c)with instantaneous return $\begin{matrix}{{\frac{{dx}_{c}}{x_{c}} = {\sum\limits_{i = 1}^{n - 1}{\alpha_{i}\frac{{dx}_{i}}{x_{i}}}}},} & (14)\end{matrix}$

[0083] x_(c) can be used as the market representative x_(m).

[0084] In practice, a most-correlated asset is obtained by consideringonly those securities that are obviously closely related to theunderlying variable x_(e).

[0085] In the case of grapefruit, for example, a most-correlated assetmight be a futures contract on orange juice. (There are no grapefruitfutures.) Such contracts are marketed securities, and are likely to behighly correlated with grapefruit prices. As another example, in seekinga most-correlated asset for DRAM (computer memory chips), one mightconsider the stock of several companies that produce DRAM and then findthe combination of these stocks that is most correlated with DRAMprices.

[0086] Ideally, this step entails identification of suitable candidates,evaluation of the co-variances of these candidates among themselves andwith the underlying variable, and optimization of the combination ofthese candidates to provide the highest correlation. These steps are notdifficult provided one has historical data on the security prices andthe underlying variable.

[0087] However, it should be understood that correlations, variances,and drift rates are measured approximately, and hence in practice exactparameters are not known, even for the standard Black-Scholes equation.Likewise, it is impossible to prove that a given asset is, in fact, theasset that is ‘most’ correlated with the under-lying variable. As in anymethod driven by estimates from limited data, one attempts to balanceaccuracy with data gathering and processing effort. Thus, in the contextof the present description, a ‘most correlated’ asset or variable isdefined to include not only the maximally correlated asset or variable,but also approximations thereto.

[0088] Although in practice a ‘most-correlated’ asset is not known withcertainty, this embodiment is distinguished from the following one bythe fact that the search for the strongest correlation is restricted toobvious candidates within one or a few related market sectors orindustries. For example, the steel industry likely would not beconsidered as a candidate for an asset of grapefruit production. Thiscontrasts with the Markowitz portfolio (generally approximated by themarket portfolio) which includes all marketed securities.

[0089] (b) The Markowitz or market portfolio. This alternativeembodiment produces the same price as the method described above. Thismethod is typically less convenient than the first and does not lead toan optimal hedging (or, equivalently, replicating) portfolio. In theory,this alternative is determined by solving the system of equations${\sum\limits_{j = 1}^{n - 1}{\sigma_{ij}\alpha_{j}}} = {\mu_{j} - {r.}}$

[0090] These α_(i)'s are then normalized to sum to 1 and used to definethe portfolio with instantaneous return$\frac{{dx}_{M}}{x_{M}} = {{\sum\limits_{i = 1}^{n - 1}\frac{{dx}_{i}}{x_{i}}} \equiv {{\mu_{M}{dt}} + {\sigma_{M}{{dz}_{M}.}}}}$

[0091] Then x_(M) can be used as the market representative x_(m).

[0092] As an approximation to the Markowitz portfolio, the marketportfolio consisting of the capitalized weighted average of the marketedassets may be used. For example, a diversified mutual fund can be used,or the S&P 500 index. Each of these is nevertheless only anapproximation. It would be essentially impossible to determine the trueMarkowitz portfolio because of the inherent error associated withdetermining the required parameters.

[0093] In contrast to the previous embodiment, the steel sector, andindeed all sectors of the economy, would be represented in this method.The advantage of this method is that the Markowitz (or market)portfolio, once found, can be used for derivatives of any underlyingvariable.

[0094] (c) Local index. Suppose there are two non-marketed variablesx_(e1) and x_(e2) that determine various derivative assets. Let x_(c1)and x_(c2) be corresponding most-correlated risky marketed assets. Nextlet x_(m) be the Markowitz combination of x_(c1) and X_(c2). Then thesingle asset x_(m) can be used as a market asset for all derivatives ofx_(e1) and x_(e2). This procedure can be directly extended to any numberof underlying variables.

[0095] To complete the set up, the payoff function F(x_(e)(T)), theparameters μ_(e) and σ_(e) of the underlying variable, the parametersμ_(m), β_(em), and the interest rate r are specified. As stated earlier,in practice it is not necessary to specify the parameters of everymarketed security. A market representative can be determined byconsideration of those assets closely related to the underlying variableor, alternatively, to a broad index of the market.

[0096] 3. Define the Differential Equation. In step 220 various termsare specified for the extended Black-Scholes equation: $\begin{matrix}\begin{matrix}{{{rV}\left( {x_{e},t} \right)} = {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}{x_{e}\left\lbrack {\mu_{e} - {\beta_{em}\left( {\mu_{m} - r} \right)}} \right\rbrack}} +}} \\{{{\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}x_{e}^{2}\sigma_{e}^{2}},}}\end{matrix} & (15)\end{matrix}$

[0097] where

ρ_(em)=E[dz_(e)dz_(m)]

β_(em)=ρ_(em)σ_(e)/σ_(m).

[0098] The subscript m denotes the market representative. The boundaryconditions are

V(x_(e), T)=F(x_(e)(T))

V(0, t)=e ^(−r(T−t)) F(0).

[0099] Equation (15) plays a central role in several embodiments of theinvention. It differs from the standard Black-Scholes equation in thecoefficient [μ_(e)−β_(em)(μ_(m)−r) This coefficient replaces r in thestandard Black-Scholes equation as a coefficient of V_(x) _(e) (x_(e),t)x_(e). The new coefficient uses properties of the marketrepresentative x_(m). Equation (15) reduces to the Black-Scholesequation if the new asset is a pure derivative so that x_(m)=x_(e) isthe most-correlated market representative.

[0100] 4. Solve the Differential Equation. In step 230 the differentialequation is solved. This can be done analytically in some cases. Mostoften it will be solved by standard numerical techniques. Once theparameters are known, solution of the equation is no more difficult thansolution of the standard Black-Scholes equation. It is only onecoefficient that is different.

[0101] The value of V(x_(e)(0), 0) is the proper price of thederivative. The procedure can be terminated here, with the proper price.Additional information is obtained in the next two steps.

[0102] 5. Find the Optimal Replicating Portfolio. Optionally, step 240calculates the best replicating portfolio. One may replicate the marketportion of the risk in the asset by holding an appropriate portfolio ofmarket securities with initial value equal to V(x_(e)(0), 0). Thisportfolio is a combination of the most-correlated asset and the riskfree asset. The amount to be held in the most-correlated asset as afunction of x_(e) and t is

φ=V_(x) _(e) (x_(e), t)x_(e)β_(ec)

[0103] where in this case, we use the subscript c to emphasize that themarket representative is taken to be the most-correlated asset. Holdingthe negative of this replicating portfolio produces an optimal hedge forthe derivative. This procedure is essentially identical to that used instandard derivatives of the Black-Scholes type. The difference is thatin the Black-Scholes case the hedge is perfect, while in embodiments ofthis invention the hedge may not be perfect.

[0104] For example, in the case of grapefruit production, the optimalhedge would be a portfolio consisting of the risk free asset and aposition in orange juice futures. The proportions of the two componentsof the portfolio would be adjusted frequently according to the formulafor φ. If the hedge is in place, the hedge portfolio yields cashimmediately equal to V(x_(e)(0), 0) and the final net payoff (consistingof sale of the grapefruit production and settling of the hedge contract)is on average zero, but it has a residual variance.

[0105] Ideally, the portfolio adjustment would be continuous, but inpractice periodic adjustments (daily, weekly, or monthly) aresufficiently accurate. These types of adjustments are common in purederivatives methodology as well, where the ideal hedge is perfect.

[0106] 6. Find the Error Variance. Optionally, step 250 calculates thetracking error. The variance of the part of the payoff that cannot behedged is found by solving another partial differential equation whichuses the solution to (15) as an input. The equation is $\begin{matrix}{{{S_{t}\left( {x_{e},t} \right)} + {S_{x_{e}}\mu_{e}x_{e}} + {\frac{1}{2}S_{x_{e}x_{e}}\sigma_{e}^{2}x_{e}^{2}} + {{^{2{r{({T - t})}}}\left\lbrack {V_{x_{e}}\sigma_{e}x_{e}} \right\rbrack}^{2}\left( {1 - \rho_{ec}^{2}} \right)}} = 0} & (16)\end{matrix}$

[0107] with boundary condition S(x_(e), T)=0. The value of S(x_(e)(0),0) is the variance of the error at time T as seen at time 0. This may beextended to computation of the variance associated with combinations ofderivatives.

[0108] For example, in the case of grapefruit, the value of S(x_(e)(0),0) gives the variance of the exposure at time T with an optimal orangejuice futures hedge. That is, with the hedge, V(x_(e)(0), 0) is obtainedat t=0 and the net amount (of the combined grapefruit production andhedge) attained at t=T has expected value 0 but variance equal toS(x_(e)(0), 0).

[0109] This calculation of error variance has no analogy in the standardBlack-Scholes case because in that case the error is always zero (intheory). Hence, this method for finding the hedge variance is new andimportant. It is not difficult to solve the appropriate partialdifferential equation, for indeed, it is similar in structure to theequation for value.

[0110] Extensions

[0111] There are several important extensions of the method. Forexample,

[0112] 1. Inclusion of cash flows that occur with time, with incrementalcash at time t being of the form h(x_(e), t)dt.

[0113] 2. Varying parameters. The parameters of the model can vary witht, x_(e), and the x_(i)'s. In general this leads to a partialdifferential equation of higher order.

[0114] 3. Parameters, such as the risk free rate, may be governed bystochastic processes.

[0115] 4. Additional variables. Additional non-marketed variables can beintroduced. These serve as “state variables” for the system. Forexample, a state variable might be total industry productive capacity,or estimates on the probability that certain legislation will be passed.

[0116] 5. Estimation variables. A suitable non-marketed but observedvariable may be the estimate of an unobserved variable that serves asthe underlying variable for the payoff. If the estimate converges to theactual value at the time of payoff, the estimate may be used at allpoints instead of the original variable. For example, the best estimatefor yearly revenue may converge to the actual figure as the year ends.

[0117] 6. Non-market random components. If the final payoff is afunction of the form F*(x_(e), y, T) where y is random and independentof x_(e) and the market, then we define F(x_(e), t)=E_(y)[F*(x_(e), y,T)] where E_(y) denotes expectation with respect to y. The value of thepayoff can then be found with this F and it will be the proper price. Inthis case, the boundary condition for auxiliary differential equationfor variance is S(x_(e), T)=variance _(y)[F*(x_(e), y, T)].

[0118] 7. Alternative Processes. The method of relating the non-marketvariable to the market in order to obtain the correct price can beextended to alternative processes, including jump processes, and movingaverage processes.

[0119] 8. Market Cash Flows. In some cases the cash flows or payoffs maydepend on marketed variables as well as non-marketed variables. In sucha case the value function V will depend on both x_(e) and the marketedvariables. The most-correlated asset for the marketed variables will bethose variables themselves, while the market representative for thenon-marketed variable is found as in the basic case.

[0120] 9. Path Dependent Cash Flows. These are cash flows such asmax_(t)[0, x_(e)(t)], 0≦t≦T that depend on the actual path taken byx_(e) rather than on its instantaneous or final value. The risk-neutralprocess can be used to evaluate such situations by taking therisk-neutral discounted expected value of all cash flows.

[0121] Brief Derivation of the Equations

[0122] We refer to step #1 (Set up) to define the problem. Hence, wehave an underlying variable x_(e) governed by geometric Brownian motion(GBM) over 0≦t≦T as

dx _(e)=μ_(e) x _(e) dt+σ _(e) x _(e) dz _(e),   (17)

[0123] and likewise there are n marketed assets that also are governedby GBM. There is a special asset of interest whose payoff at time T isF(x_(e)(T)).

[0124] We propose an (initially unknown) price function V(x_(e), t) onthe time interval [0, T] with terminal value V(x_(e), T)=F(x_(e)). Itwill be defined by instantaneous discounted projection.

[0125] At a fixed time t we define the space M as the linear spacegenerated by all instantaneous marketed returns (dx_(i))/x_(i). That is,M is made up of all linear combinations of those instantaneous returns.Symbolically, the value of our V(x_(e), t) will satisfy

V(x _(e) , t)=P{V(x _(e) , t)+dV(x _(e) , t)|M}.   (18)

[0126] which denotes the discounted projection P onto the space M ofinstantaneous marketed returns. Specifically, V(x_(e), t) is found by:first calculating the payoff m* at time t+dt in M that is closest toV(x_(e), t)+dV(x_(e), t) in the sense of minimizing the expected squareof their difference; and second, computing the price at time t of m*(which is defined by linearity since it is a combination of marketedreturns). Briefly, we say that the value at t of the payoff at t+dt isgiven by discounted instantaneous projection. This process of projectiononto M is illustrated schematically in FIG. 3.

[0127] Since payoff V(x_(e), t) at t+dt is certain, its price at t isthe discounted value (1−rdt)V(x_(e), t). Thus (18) becomes

rV(x_(e), t)dt=P{dV(x_(e), t)|M}.   (19)

[0128] We shall only keep terms that are first-order in dt. Hence, inparticular, P{dt}=dt. Substituting the Ito formula $\begin{matrix}{{{dV}\left( {x_{e},t} \right)} = {{\left\lbrack {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}\mu_{e}x_{e}} + {\frac{1}{2}V_{x_{e}x_{e}}\sigma_{e}^{2}x_{e}^{2}}} \right\rbrack {dt}} + {V_{x_{e}}\sigma_{e}x_{e}{dz}_{e}}}} & (20)\end{matrix}$

[0129] into (19) and keeping only first-order terms in dt, (19) becomes$\begin{matrix}\begin{matrix}{{{{rV}\left( {x_{e},t} \right)}{dt}} = {{\left\lbrack {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}\mu_{e}x_{e}} + {\frac{1}{2}V_{x_{e}x_{e}}\sigma_{e}^{2}x_{e}^{2}}} \right\rbrack {dt}} +}} \\{{V_{x_{e}}\sigma_{e}x_{e}P{\left\{ {dz}_{e} \middle| M \right\}.}}}\end{matrix} & (21)\end{matrix}$

[0130] The projection of dz_(e) is a linear combination of r dt and thedifferential of a marketed asset most correlated with dx_(e). A marketedasset x_(e) with instantaneous return dx_(c)/x_(c) most correlated withdx_(e)/x_(e) satisfies dx_(c)=μ_(c)x_(c)dt+σ_(c)x_(c)dz_(c) where μ_(c)and σ_(c) are each formed as (identical) linear combinations of theμ_(i)'s and σ_(i)'s, respectively. z_(c) is a standardized Wienerprocess.

[0131] Standard methods show that the projection of dz_(e) onto M isρ_(ec)dz_(c). In addition, from

x_(c) =P{x _(c)+μ_(c) x _(c) dt+x _(c)σ_(c) dz _(c)}

[0132] it follows that $\begin{matrix}{{P\left\{ {dz}_{c} \right\}} = {\frac{\left( {r - \mu_{c}} \right)}{\sigma_{c}}{{dt}.}}} & (22)\end{matrix}$

[0133] Substituting this in (21) produces $\begin{matrix}\begin{matrix}{{{{rV}\left( {x_{e},t} \right)}{dt}} = \left\lbrack {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}\mu_{e}x_{e}} +} \right.} \\{{\left. {\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}x_{e}^{2}} \right\rbrack {dt}} + {V_{x_{e}}P\left\{ {\rho_{ec}\sigma_{e}x_{e}{dz}_{c}} \right\}}} \\{= \left\lbrack {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}{x_{e}\left\lbrack {\mu_{e} - {\beta_{ec}\left( {\mu_{c} - r} \right)}} \right\rbrack}} +} \right.} \\{{\left. {\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}x_{e}^{2}} \right\rbrack {dt}},}\end{matrix} & \quad \\{{where}\quad {we}\quad {define}} & \quad \\{{\beta_{ec} \equiv \frac{\sigma_{ec}}{\sigma_{c}^{2}}} = \frac{\rho_{ec}\sigma_{e}}{\sigma_{c}}} & (23)\end{matrix}$

[0134] Canceling the dt we have the final result $\begin{matrix}\begin{matrix}{{{rV}\left( {x_{e},t} \right)} = {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}{x_{e}\left\lbrack {\mu_{e} - {\beta_{ec}\left( {\mu_{c} - r} \right)}} \right\rbrack}} +}} \\{{\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}{x_{e}^{2}.}}}\end{matrix} & (24)\end{matrix}$

[0135] This is the basic extended Black-Scholes equation.

[0136] Once equation (24) is solved with the boundary conditionsV(x_(e), t)=F(x_(e)), V(0, t)=e^(−r(T−t))F(0), the value V(x_(e)(0), 0)is the value at time 0 of the derivative.

[0137] Market Representatives

[0138] Details of how the two basic alternative market representatives,a most-correlated asset and the Markowitz portfolio, are computed aregiven here. In practice, these detailed calculations are greatlysimplified (approximated) by use of intuition or equilibrium arguments,as mentioned below.

[0139] 1. Most-Correlated Asset.

[0140] Arrange the n marketed assets so the first n−1 are risky and then-th is risk free. The return of a most-correlated asset is of the form$\begin{matrix}{{\frac{{dx}_{c}}{x_{c}} = {\sum\limits_{i = 1}^{n - 1}{\alpha_{i}\frac{{dx}_{i}}{x_{i}}}}},} & (25)\end{matrix}$

[0141] where the α_(i)'s do not necessarily sum to 1.

[0142] To maximize the correlation of this return with dx_(e), theα_(i)'s solve$\max \quad {E\left( {\frac{{dx}_{c}}{x_{c}}\frac{{dx}_{e}}{x_{e}}} \right)}$${{subject}\quad {to}\quad {{var}\left( \frac{{dx}_{c}}{x_{c}} \right)}} \leq 1.$

[0143] Introducing a (positive) Lagrange multiplier for the constraint,and representing dx_(c)/x_(c) in terms of the α_(i)'s, this isequivalent to solving $\begin{matrix}{{\max \left\lbrack {{\sum\limits_{i = 1}^{n - 1}{\alpha_{i}\sigma_{ie}}} - {\lambda {\sum\limits_{ij}^{n - 1}{\alpha_{i}\sigma_{ij}\alpha_{j}}}}} \right\rbrack}.} & (26)\end{matrix}$

[0144] Since any solution can be scaled by an arbitrary positiveconstant and still preserve the property of being most correlated,without loss of generality one may set λ=1. Then (26) becomes$\begin{matrix}{{\sum\limits_{j = 1}^{n - 1}{\sigma_{ij}\alpha_{j}}} = \sigma_{ei}} & (27)\end{matrix}$

[0145] which is easily solved.

[0146] The current price (at t) of dx_(c)/x_(c) is$\sum\limits_{i = 1}^{n - 1}{\alpha_{i}.}$

[0147] To make the price equal to 1, the α_(i)'s are scaled so that theydo sum to 1. Then the correct x_(c) is given by (25) with the scaledα_(i)'s.

[0148] In practice, a market representative may be chosen as an assetthat is obviously closely related to the variable x_(e). For example, anorange juice future is closely related to grapefruit price. It isprobably not necessary to consider oil and metal futures or thethousands of marketed stocks to find a closely related marketrepresentative. A refined approach is to consider a family ofagricultural futures and, based on historical returns, compute themost-correlated portfolio from this family. The selection of a closelyrelated market representative is, in practice, partially art andpartially data collection and computation.

[0149] 2. Markowitz portfolio. The Markowitz portfolio x_(M) has theadvantage that it can be used to price a derivative of any processx_(e), but its disadvantages are that is difficult to use, lessintuitive, may not exist, and does not lead to an optimal replication.(The use of this portfolio in the single-period case is described inLuenberger [10].) The Markowitz portfolio x_(M) is the portfolio ofpurely risky assets (the first n−1 assets) that has price 1 andmaximizes $\frac{\mu_{M} - r}{\sigma_{M}}.$

[0150] If it exists, the set of α_(j)'s that achieves it satisfies$\begin{matrix}{{\sum\limits_{i = 1}^{n - 1}{\sigma_{ij}\alpha_{j}}} = {c\left( {\mu_{i} - r} \right)}} & (28)\end{matrix}$

[0151] for the constant c that makes the α_(i)'s sum to 1. Thisportfolio is an alternative for purposes of valuation only, not fordetermining optimal replication or hedging. In practice, a broad marketindex, such as the S&P 500, is used as a proxy for the Markowitzportfolio.

[0152] 3. Local index. Suppose x_(e1) and x_(e2) are non-marketedunderlying variables and x_(c1) and x_(c2) are the respectivemost-correlated marketed assets. The Markowitz combination of x_(c1) andx_(c2) is the portfolio x_(m)=α₁x_(c1)=α₂x_(c2) where the weights α₁ andα₂ are determined as in the above discussion of the Markowitz portfoliounder the assumption that x_(c1), x_(c2) and r are the only marketedassets. This x_(m) is a market representative for any derivative ofx_(e1) and x_(e2). It serves as a local index, and may in practice bethe index of an industry to which x_(e1) and x_(e2) are related. Thislocal index can be used to value a variety of derivatives. It cannot beused as the basis for optimal hedging, but it can be used to formapproximate hedges. The method extends to any number of underlyingvariables.

[0153] Important Special Case.

[0154] If the payoff is a put or call option on the variable x_(e), aclosed-form expression for the solution to (15) applies, which extendsthe special case solved by Black and Scholes. If the strike price of thecall option is K, the payoff is max(x_(e)−K, 0]. The value of the calloption is

V(x _(e) , t)=x _(e) e ^([(ω−r)(T−t)]) N(d ₁)−Ke ^(−r(T−t)]) N(d ₂)  (29)

[0155] where N(d) denotes the value of the standard normal distributionat amount d, and $\begin{matrix}{d_{1} = \frac{{\ln \left( {x_{e}/K} \right)} + {\left( {w + {\frac{1}{2}\sigma_{e}^{2}}} \right)\left( {T - t} \right)}}{\sigma_{e}\sqrt{T - t}}} & (30) \\{d_{2} = {d_{1} - {\sigma_{e}\sqrt{T - t}}}} & (31)\end{matrix}$

 ω=μ_(e)−β_(em)(μ_(m) −r),   (32)

[0156] where μ_(m) is the drift rate of the market representative x_(m)and β_(em) is the beta of x_(e) and x_(m).

[0157] The value of a put with strike price K can be found from anextended put-call parity formula

C−P+Ke ^(−rT) =x _(e) e ^((ω−r)T).   (33)

[0158] Here C is the price of the call, and P the price of the put.

[0159] Optimal Replication

[0160] It is possible to optimally replicate the final payoff by tradingin the marketed assets. The replication is imperfect, but the error haszero expected value and is uncorrelated with all marketed assets. Thisis, therefore, the best that can be done within a mean-varianceframework. The trading strategy for replication can be derived from thevalue function.

[0161] As in (20), the value function follows the Ito process$\begin{matrix}\begin{matrix}{{{dV}\left( {x_{e},t} \right)} = {{\left\lbrack {{V_{t}\left( {x_{e},t} \right)} + {{V_{x_{e}}\left( {x_{e},t} \right)}\mu_{e}x_{e}} + {\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}x_{e}^{2}}} \right\rbrack {dt}} +}} \\{{V_{x_{e}}\sigma_{e}x_{e}{{dz}_{e}.}}}\end{matrix} & (34)\end{matrix}$

[0162] One may write $\begin{matrix}{{{d\quad z_{e}} = {{\rho_{ec}d\quad z_{c}} + {\sqrt{1 - \rho_{ec}^{2}}d\quad z_{p}}}},} & (35)\end{matrix}$

[0163] where dz_(p) is a standardized Wiener process uncorrelated withall marketed assets and whereρ_(ec)≡σ_(ec)/(σ_(e)σ_(c))=β_(ec)σ_(c)/σ_(e). With this substitution andsubstitution of the term$\frac{1}{2}{V_{x_{e}x_{e}}\left( {x_{e},t} \right)}\sigma_{e}^{2}x_{e}^{2}$

[0164] from the extended Black-Scholes equation (15) it is possible totransform (34) to

dV(x _(e) , t)=[(V(x _(e) , t)−φ)r+φμ _(c) ]dt+φσ _(c) dz _(c) +δdz_(p),   (36)

[0165] where

φ(x _(e) , t)=V _(x)(x _(e) , t)x _(e)β_(ec)   (37)

δ(x _(e) , t)=V _(x) _(e) (x _(e) , t)σ_(e) x _(e){square root}{squareroot over (1−ρ_(ec) ²)}.   (38)

[0166] Motivated by (36), the process H is defined by

dH(x _(e) , t)=[(H(x _(e) , t)−φ)r+φμ _(c) ]dt+φσ _(c) dz _(c),   (39)

[0167] with initial condition H(x_(e), 0)=V(x_(e), 0).

[0168] From (39) it is clear that dH(x_(e), t) is in M at (x_(e), t) andthat, overall, H(x_(e), t) is generated by linear combinations ofmarketed returns. From

d(V−H)=r(V−H)dt+δdz _(p).   (40)

[0169] and the initial condition V−H=0, it is clear that at time 0 thereholds E[V(x_(e), t)−H(x_(e), t)]=0 for all t, 0≦t≦T. Furthermore, sinceV−H is a linear integral of dz_(p)'s, it is orthogonal (that is,uncorrelated) with all marketed returns. Thus the random variableH(x_(e), T) as seen at time 0 is the projection of V(x_(e), T) onto thespace of marketed assets over 0≦t≦T. It follows that the optimalreplication is simply H, governed by (39). This replication requiresonly initial cash of V(x_(e), 0) and no other infusions or withdrawals.At each instant H−φ is invested in the risk free asset and φ in themost-correlated asset (for a total of H). Holding −H serves to optimallyhedge the derivative.

[0170] Projection Error

[0171] The variance of the residual replication error V−H at T can befound as a solution to a partial differential equation adjunct to thegeneral pricing equation.

[0172] Define the difference variable D=V−H, and let U=e^(r(T−t))D. Thenfrom (40) $\begin{matrix}\begin{matrix}{{dU} = {{{- {re}^{r{({T - t})}}}{DdT}} + {e^{r{({T - t})}}{dD}}}} \\{= {{\left\{ {{{- {re}^{r{({T - t})}}}D} + {e^{r{({T - t})}}{rD}}} \right\} {dt}} + {e^{r{({T - t})}}{\delta \left( {x_{e},t} \right)}{dz}_{p}}}} \\{{= {{\delta^{*}\left( {x_{e},t} \right)}{dz}_{p}}},}\end{matrix} & (41)\end{matrix}$

[0173] where δ*(x_(e), t)=e^(r(T−t))δ(x_(e), t).

[0174] Let S(x_(e), t) be the variance of U(T) when in (41) U isinitiated with U=0 at the point (x_(e), t). This variance is$\begin{matrix}{{{S\left( {x_{e},t} \right)} = {E\left\lbrack {\int_{t}^{T}{{\delta^{*}\left( {{x_{e}(s)},s} \right)}^{2}{s}}} \right\rbrack}},} & (42)\end{matrix}$

[0175] where E denotes expectation at t.

[0176] We write (42) as $\begin{matrix}\begin{matrix}{{S\left( {x_{e},t} \right)} = {{{\delta^{*}\left( {x_{e},t} \right)}^{2}{dt}} + {E\left\lbrack {\int_{t + {dt}}^{T}{{\delta^{*}\left( {{x_{e}(s)},s} \right)}^{2}{s}}} \right\rbrack}}} \\{= {{{\delta^{*}\left( {x_{e},t} \right)}^{2}{dt}} + {{E\left\lbrack {{S\left( {x_{e},t} \right)} + {{dS}\left( {x_{e},t} \right)}} \right\rbrack}.}}}\end{matrix} & (43)\end{matrix}$

[0177] Hence,

E[dS(x _(e) , t)]+δ*(x _(e) , t)² dt=0.   (44)

[0178] Expanding (44) using Ito's lemma, we find $\begin{matrix}{{{{S_{t}\left( {x_{e},t} \right)} + {S_{x_{e}}\mu_{e}x_{e}} + {\frac{1}{2}S_{x_{e}x_{e}}\sigma_{e}^{2}x_{e}^{2}} + {{e^{2{r{({T - t})}}}\left\lbrack {{V_{x_{e}}\left( {x_{e},t} \right)}\sigma_{e}x_{e}} \right\rbrack}^{2}\left( {1 - \rho_{ec}^{2}} \right)}} = 0},} & (45)\end{matrix}$

[0179] with boundary condition S(x_(e), T)=0. The value S(x_(e), 0) isthe variance of the replication error at T, as seen at t=0.

[0180] Universality

[0181] As a by-product of the optimal replication equations, we obtain acompelling justification for methods used in embodiments of thisinvention. When the new asset is priced according to teachings of thisinvention, the optimal replication H has the same expected value as theasset with payoff F(x_(e)(T)). In addition the optimal replication haslower variance than the asset since V(x_(e), T)=H(x_(e), T)+error, wherethe error is uncorrelated with all marketed assets. This means thatevery risk-averse investor will prefer the replication over the newasset. This property is termed universality, to reflect that it is truefor everyone. Another way to state this property is that if the newasset is priced according to methods of this invention, everyrisk-averse investor will elect to include the asset only at the zerolevel (preferring neither to be long or short the asset). In this sense,the price renders the asset irrelevant in the market, for it isdominated by the best replicating asset which is already available.

[0182] Risk-Neutral Process.

[0183] These methods can be transformed to formulas based on arisk-neutral processes. The novel feature is that the risk-neutralprocess is defined in terms of a market representative x_(m) definedearlier. The appropriate risk-neutral process is

dx _(e) =ωx _(e) dt+σ _(e) x _(e) dz _(e)   (46)

[0184] where

ω=μ_(e)−β_(em)(μ_(m) −r).   (47)

[0185] In these terms the proper price of the payoff F(x_(e)(T)) is

p=e ^(−rT) Ê[F(x _(e)(T))]  (48)

[0186] where Ê denotes expectation at time 0 with respect to therisk-neutral process. One way to solve such a problem is by simulation.Many paths of x_(e) are generated according to the risk-neutral processand the resulting terminal payoffs are averaged to obtain an estimate ofthe risk-neutral expected value. The discounted value of this estimateconverges to the correct price as the number of simulation runsincreases.

[0187] A significant advantage of the risk-neutral process is that itcan be used to value path-dependent cash flows; where the payoff dependson the particular path taken by x_(e) as well as its final value. Forexample, the Asian option with payoff max [max_(0≦t≦T)x_(e)(t)−K, 0] canbe evaluated using the risk-neutral process. The proper price is again

p=e ^(−rT) Ê[F(x _(e)(T)],

[0188] which can be evaluated most easily by simulation, but also byspecial grid methods.

DETAILED DESCRIPTION OF THE INVENTION:

[0189] Discrete-Time Case.

[0190] Approximation

[0191] One way to work in a discrete-time framework is to directlydiscretize the model and the partial differential equations. Forexample, the new version of the process for x_(e) becomes

x _(e)(k+1)=(1+μ_(e) Δt)x _(e)+σ_(e) x _(e)∉(k){square root}{square rootover (Δt)}.   (49)

[0192] In this equation Δt is the length of the time step of the modeland ∉(k) is the value of a standardized normal random variableuncorrelated with previous or future such variables. The marketvariables are transformed to discrete form in a similar way. Acorresponding discrete-time version of the extended Black-Scholesequation can be developed. It is $\begin{matrix}{{V_{k - 1}\left( {x_{e}\left( {k - 1} \right)} \right)} = {\frac{1}{R}\left\{ {E\left\lbrack {{V_{k}\left( {x_{e}(k)} \right\rbrack} - \frac{{cov}\left( {{V_{k}\left( {x_{e}(k)} \right)}\left( {{E\left\lbrack {x_{m}(k)} \right\rbrack} - {Rp}_{m}} \right)} \right.}{\sigma_{m}^{2}}} \right\}} \right.}} & (50)\end{matrix}$

[0193] where R=e^(tΔt) and where x_(m) is the market representative(either the Markowitz portfolio or a most-correlated market asset or acombination of these.) The expected values are taken at time k−1. Thequantity p_(m) is the price at time k−1 of x_(m), σ_(m) ² is thevariance of x_(m)(k) as seen at k−1. The formula is valid for anydiscrete process that governs x_(e), although the market variablesfollows a fixed process as in (49) (but with i's instead of e assubscripts). The extensions applicable in the continuous-time frameworkare also applicable in discrete time.

[0194] The discrete-time approximation (49) can be converted torisk-neutral form as

x _(e)(k+1)=(1+ωΔt)x _(e)+σ_(e) x _(e)∉(k){square root}{square root over(Δt)},   (51)

[0195] where ω=μ_(e)−β_(em)(μ_(m)−r).

[0196] Finite-State Models

[0197] A single stochastic variable A is represented by the two-statemodel shown in FIG. 4. The U_(A) outcome is considered an “up” move, andthe D_(A) outcome a “down” move. The model has three degrees of freedom(one probability and two node values) and hence the expected value andvariance of the variable can be matched with one remaining degree offreedom. The model can be extended over several time periods as abinomial tree or (frequently) a binomial lattice.

[0198] There are two standard approaches to parameter matching: additiveand geometric, corresponding to matching moments of the variable itselfor matching moments of the logarithm of the payoff. Geometric matchingis a natural choice for processes governed by geometric Brownian motion.Additive matching is frequently used for discrete-time models. Abinomial model for one of these is easily converted to a correspondingbinomial model for the other. For small Δt defining the time step of themodel, the two approaches are nearly identical.

[0199] Suppose that A and B are variables as above, and suppose G is afunction of (a derivative of) B. G is defined by the two values G_(u)and G_(d) corresponding to whether U_(B) or D_(B) occurs. There is animportant relation between the covariance of such a derivative with Aand the covariance of B with A that holds when B is described by abinomial model. This is spelled out in the following result, easilyproved by algebra: $\begin{matrix}{{{cov}\left( {A,G} \right)} = {\frac{\left( {G_{u} - G_{d}} \right)}{\left( {U_{B} - D_{B}} \right)}{{{cov}\left( {A,B} \right)}.}}} & (52)\end{matrix}$

[0200] Suppose A is the market asset most correlated with B. The aboveresult guarantees that in the case of binomial models, A is alsomost-correlated to all derivatives of B (since the two covariances areproportional). It follows that the three-variable model (with A, B, andthe risk free total return R) can be used to price all derivatives of B.

[0201] Paralleling the development for the continuous-time case, A is amarket representative and it may be taken to be a marketed assetmost-correlated with the underlying B or, alternatively, as theMarkowitz (market) portfolio.

[0202] Let G be a derivative of B, defined by its two values G_(u) andG_(d). The projection price is (with an over-bar on a random variabledenoting expected value) $\begin{matrix}\begin{matrix}{\upsilon_{G} = {\frac{1}{R}\left\lbrack {{E(G)} - {{{cov}\left( {G,A} \right)}{\left( {\overset{\_}{A} - {\upsilon_{A}R}} \right)/\sigma_{A}^{2}}}} \right\rbrack}} \\{= {\frac{1}{R}\left\lbrack {{p_{B}G_{u}} + {\left( {1 - p_{B}} \right)G_{d}} - {{\left\lbrack {{{{cov}\left( {{1\left( U_{B} \right)},A} \right)}G_{u}} + {{{cov}\left( {{1\left( D_{B} \right)},A} \right)}G_{d}}} \right\rbrack \left\lbrack {\overset{\_}{A} - {\upsilon_{A}R}} \right\rbrack}/\sigma_{A}^{2}}} \right\rbrack}} \\{= {\frac{1}{R}\left\{ {\left\lbrack {p_{B} - {{\beta_{{1{(U_{B})}},A}\left\lbrack {\overset{\_}{A} - {\upsilon_{A}R}} \right\rbrack}G_{u}} + {\left\lbrack {\left( {1 - p_{B}} \right) - {\beta_{1{(D_{B}}}\left\lbrack {\overset{\_}{A} - {\upsilon_{A}R}} \right\rbrack}} \right\rbrack G_{d}}} \right\},} \right.}}\end{matrix} & (53)\end{matrix}$

[0203] where 1(U_(B)) and 1(D_(B)) denote payoffs of 1 if U_(B) orD_(B), respectively, occurs; and where

β_(1(U) _(B),A) =cov[1(U_(B)), A]/σ_(A) ².

[0204] This can be written as $\begin{matrix}{V_{G} = {\frac{1}{R}\left\lbrack {{q_{B}G_{u}} + {\left( {1 - q_{B}} \right)G_{d}}} \right\rbrack}} & (54)\end{matrix}$

[0205] where

q _(B) =p _(B)−β_(1(U) _(B) _(),A) [{overscore (A)}−ν _(A) R].   (55)

[0206] Expanding the beta term, we can calculate explicitly

q _(B) =p _(B) −p _(B)(1−p _(B)) [A _(u) −A _(d) ][{overscore (A)}−ν_(A) R]/σ _(A) ²,   (56)

[0207] where A_(u)=E[A|U_(B)] A_(d)=E[A|D_(B)].

[0208] This is the basic method for the discrete-time case. The uniquefeature is the formula (55) or (56) for the risk-neutral probability ofan up move. Once this probability is determined, it may be used in placeof the true probability p_(A) for the purpose of evaluating payoffs thatdepend on B. In other words, for purposes of evaluation the binomialmodel takes the form of FIG. 5.

[0209] Recursive Solution

[0210] The single-period structure can be extended to a multiple-periodframework by piecing together single periods, and this leads to arecursive solution.

[0211] Suppose the nodes of the lattice are numbered by the time index kand the state index s_(k) which is the level of the node counting fromthe bottom. Briefly, we write the node as (k, s_(k)). The value functionof the derivative is a value at each node, and described as V_(k)(s_(k))at time point k and state s_(k).

[0212] The recursive solution is $\begin{matrix}{{{V_{k - 1}\left( s_{k - 1} \right)} = {\frac{1}{R}\left\lbrack {{q_{B}{V_{k}\left( s_{k - 1}^{u} \right)}} + {\left( {1 - q_{B}} \right){V_{k}\left( s_{k - 1}^{d} \right)}}} \right\rbrack}},} & (57)\end{matrix}$

[0213] where s_(k−1) ^(u) denotes the upper successor state to s_(k−1)and s_(k−1) ^(d) denotes the lower successor state to s_(k−1). Theprocess is started with the terminal boundary condition specifying thepayoff of the derivative G. If there are additional payoffs along theway, they are incorporated step by step in the usual manner.

[0214] This method can be extended in the same ways as thecontinuous-time version.

[0215]FIG. 6 is a schematic depiction of the method for thediscrete-time case. This schematic has the same basic structure as thatof FIG. 1 which depicts the schematic for the continuous-time case, withthe exception that the variable V is found through a recursion process600. The figure shows how the underlying variable B defines the newasset and how the market representative A is extracted from the market.The properties of these two variables define q_(B) which defines therecursion, leading to the value of the new asset.

[0216] Error Process

[0217] Denote by {V_(k)|M} the projection of V_(k) on the market at timek−1. This will be the projection of V_(k) onto the space spanned by Rand a market representative A most correlated with B. It is easily shownthat

{V _(k) |M}={overscore (V)} _(k) +cov(V _(k) , A)(A−{overscore(A)})/σ_(A) ².

[0218] Hence

V _(k) ={overscore (V)} _(k) +cov(V _(k) , A)(A−{overscore (A)})/σ_(A)²+∉_(k),   (58)

[0219] where ∉_(k) is uncorrelated with the market. Using the pricingequation (50), one has

RV _(k−1) ={overscore (V)} _(k) +cov(V _(k) , A)(R−{overscore(A)})/σ_(A) ².

[0220] Eliminating {overscore (V)}_(k), (58) can be written as$\begin{matrix}\begin{matrix}{V_{k} = {{RV}_{k - 1} - {{{cov}\left( {V_{k},A} \right)}{\left( {R - \overset{\_}{A}} \right)/\sigma_{A}^{2}}} + {{{cov}\left( {V_{k},A} \right)}{\left( {A - \overset{\_}{A}} \right)/\sigma_{A}^{2}}} + \varepsilon_{k}}} \\{= {{\left\lbrack {{\left( {1 - \gamma} \right)R} + {\gamma \quad A}} \right\rbrack V_{k - 1}} + \varepsilon_{k}}}\end{matrix} & (59)\end{matrix}$

[0221] where

γ=cov(V _(k) /V _(k−1) , A)/σ_(A) ².

[0222] In the same manner as in the continuous-time case this shows howto select the best approximating (or replicating) portfolio H, with anamount H_(k−1)−γV_(k−1) in the risk free asset and γV_(k−1) in theportfolio A (for a total of H_(k−1)). Thus H satisfies the recursion

H _(k)=(H _(k−1) −γV _(k−1))R+γV _(k−1) A+∉ _(k).   (60)

[0223] Error Propagation

[0224] The replication error D_(k)≡V_(k)−H_(k) satisfies

D_(k) =RD _(k)+∉_(k)   (61)

[0225] where ∉_(k) is uncorrelated with the market and

∉_(k) =V _(k) −{overscore (V)} _(k) −cov(V _(k) , A)(A−{overscore(A)})/σ_(A) ².   (62)

[0226] It follows that

var(∉_(k))=var(V _(k))−cov(V _(k) , A)²/σ_(A) ².   (63)

[0227] In terms of the lattice parameters this becomes $\begin{matrix}{{{var}\left( \varepsilon_{k} \right)} = {{\left\{ {{p_{A}\left( {1 - p_{A}} \right)} - \frac{{cov}\quad \left( {A,B} \right)^{2}}{\left( {U_{B} - D_{B}} \right)^{2}\sigma_{A}^{2}}} \right\} \left\lbrack {V_{k}^{u} - V_{k}^{d}} \right\rbrack}^{2}.}} & (64)\end{matrix}$

[0228] If the dynamics of the variables A and B are stationary, theexpression in brackets is constant, and hence the error variance issimply a constant times the square of the difference in the twosuccessor V_(k)'s.

[0229] Let U_(k)=R^(T−k)D_(k). Then U_(k)=U_(k−1)+R^(T−k)∉_(k). If S_(k)is the variance of U_(T) as seen starting at k, then

S _(k−1) =E[S _(k) ]+R ^(2(T−k)) var(∉_(k))   (65)

[0230] with terminal condition S_(T)=0. This can now be evaluated by abackward recursion in the lattice.

[0231] Method Steps

[0232] Following is a summary of the steps of the lattice method for thediscrete-time case. These steps are depicted in FIG. 7 and parallelthose for the continuous-time case illustrated in FIG. 2.

[0233] 1. Step 700 is the set up. Given a random payoff G that dependson a non-traded variable B that evolves randomly, formulate a binomiallattice model.

[0234] 2. Step 710 determines an appropriate market representative A aseither (a) a market asset most correlated to B or (b) the Markowitz ormarket portfolio.

[0235] 3. Step 720 determines the risk-neutral probabilities by usingthe formula (56).

[0236] 4. Step 730 solves for the values of V on the latticecorresponding to the variable B using the recursion (57). Initiate withthe terminal boundary condition V=G. The result at time 0 is the price.

[0237] 5. Optionally, step 740 determines the optimal replicatingportfolio from H₀=V₀, and thereafter at step k−1 investingH_(k−1)−γV_(k−1) in the risk free asset and γV_(k−1) in the asset mostcorrelated with B.

[0238] 6. Optionally, step 700 determines the error variance by thebackward recursion (65).

[0239] Implementation

[0240] The present invention may be implemented in various differentways on a computer. For the purposes of the present description, theterm ‘computer’ is defined to include any electronic digital informationprocessor, including hand-held calculators, personal data assistants,pocket personal computers, laptop computers, desktop computers, and soon. The computer may be programmed by providing a computer readabledigital storage medium containing instructions to execute variousmethods of the invention. The program or programs are then executed onthe computer. Information associated with the variable x_(m) may beprovided to the computer manually by a user, received from a datanetwork, or retrieved from a storage medium into computer memory. Theprogram then retrieves the information associated with the variablex_(m) from the computer memory and performs calculations in accordancewith various methods of the invention. The results of the calculationsmay be used to display or otherwise communicate information to the user,or they may be stored or transmitted digitally for further processing orlater display.

[0241] More specifically, the implementation of methods of the inventionmay take various different forms. Following are representative examplesof specific implementations.

[0242] 1. Numerical Solution. The straightforward way to implement thecontinuous-time method once the model is defined is by solution of theextended Black-Scholes equation (15). This can be carried out bystandard numerical procedures, following the methods developed for theordinary Black-Scholes equation. The simple, one-dimensional versionshown explicitly in (15) can be solved by a finite grid method: a largetwo-dimensional grid of points is defined with coordinates correspondingto x_(e) and t and grid point values corresponding to the associated Vvalues. In the simplest method, first-order derivatives of V are formedas (normalized) differences in grid point values. The second-orderderivative is the second-order difference of grid point values. Boundaryconditions are typically imposed at t=T, and at lower and upper valuesof x_(e). In practice, if the problem is defined for all values of x_(e)with 0≦x_(e)≦∞, an appropriate value for V(x_(e), t) at a high upperboundary curve {overscore (x)}_(e)(t) is assigned as a boundarycondition in addition to the terminal condition and the condition atx_(e)=0. It is possible to carry out the complete solution with aspreadsheet program such as Excel running on a general purpose computer.Experience has shown that the primary technical concern for simplemethods such as this is that the time step between successive gridpoints be small in order to assure convergence. Small grid point widthsin both x and t are used to get good accuracy.

[0243] For higher-order equations associated with extensions of thebasic method, it is convenient to use a professional software package orone written in a programming language. The value of a call option on anon-tradable variable, but with a stochastically varying interest ratehas been solved by the inventor using the Excel Visual Basic package.

[0244] The same techniques may be applied to solve for the varianceaccording to the partial differential equation (45). In simpleexperiments, the same grid size as used for finding V worked for findingthe non-hedgable variance.

[0245] 2. Explicit Solution. The explicit solution for call options (orfor puts using the put-call parity formula) is easily solved by usingeither a table, or more likely, an approximate formula for values of thenormal distribution. Again this can be carried out with Excel. It couldbe easily incorporated into a hand-held calculator.

[0246] 3. Discrete-time Version. The discrete time recursion is easilycarried out by direct recursion. This method is relatively free fromconvergence issues, since both the model and the solution method arecarried out with the same time steps (unlike the continuous-time casewhich is computed with a discrete process).

[0247] 4. Lattice Methods. The lattice method is easily implemented. Insimple cases it may be carried out with a spreadsheet program running ona computer. The method becomes more challenging to implement when themodel parameters are not constant, and when higher-order lattices andtrees are used.

[0248] 5. Risk-Neutral Computation. Beginning with a discrete model, therisk-neutral version of the processes can be implemented computationallyby computing the risk-neutral expected value of the payoff function.This can be carried out with backward recursion, simulation, or byconstructing a lattice that has the risk-neutral probabilities on itsarcs and using backward recursion on that lattice.

[0249] 6. Simulation. Simulation is a powerful method for implementingmethods of this invention. Standard simulation packages can be used,such as a spreadsheet program like Excel, modest simulation packagessuch as Crystal Ball, or more advanced statistical packages. At aprofessional level, special software that accounts for the financialstructure as well as advanced simulation concepts would be used.

[0250] 7. Optimization. In many situations, on-going decisions can bemade that influence the value of the payoff. For example, anAmerican-style option allows the owner to exercise the option at anytime before expiration, and selection of this exercise policy is animportant component of the analysis of the option. Similarly, in abusiness venture there are opportunities to expand, contract, delay, andso forth. These policies can be found with the methods of this documentin conjunction with standard methods for policy optimization employedfor ordinary derivative theory.

References

[0251] [1] Sharpe, W. F. “Capital Asset Prices: A Theory of MarketEquilibrium under Conditions of Risk,” Journal of Finance, 19, 425-442,1964.

[0252] [2] Black, F., and M. Scholes, “The Pricing of Options andCorporate Liabilities,”Journal of Political Economy 81, 637-654, 1973

[0253] [3] Föllmer, H, and D. Sondermann, “Hedging of Non-RedundantContingent-Claims,” in Werner Hildenbrand and Andrew Mas-Colell, eds.,Contributions to Mathematical Economics, in Honor of Gérard Debreu,Amsterdam, North-Holland, 1986, 205-23.

[0254] [4] Davis, M. H. A. “Option Pricing in Incomplete Markets,” in M.A. H. Dempster and S. Pliska, eds, Mathematics of Derivative Securities,Cambridge University Press, Cambridge, 1997, 216-26.

[0255] [5] Bertsimas, D, L. Kogan, and A. W. Lo, “Pricing and HedgingDerivative Securities in Incomplete Markets: An ∉-Arbitrage Approach,”Massachusetts Institute of Technology working Paper #LFE-1027-97, June1997.

[0256] [6] Merton, R. C. “Applications of Options Pricing Theory:Twenty-five years later.” American Economic Review, 88 323-349. (1998)

[0257] [7] He, H, and N. D. Pearson, “Consumption and Portfolio Policieswith Incomplete Markets and Short-Sale Constraints”: The InfiniteDimensional Case,” Journal of Economic Theory, 54, 259-304, (1991)

[0258] [8] Schwartz, Eduardo S. and Mark Moon, “Rational Pricing ofInternet Companies,” Financial Analysts Journal, May/June 2000, 62-75.

[0259] [9] Luenberger, D. G. “Projection Pricing,” Journal ofOptimization Theory and Applications, April 2001, 1-25.

[0260] [10] Luenberger, D. G., “A Correlation Pricing Formula”, Journalof Economic Dynamics and Control, Jul. 26, 2002, 1113-1126.

[0261] [11] Holtan, H. M. “Asset Valuation and Optimal Portfolio Choicein Incomplete Markets,” Ph. D. Dissertation, Department ofEngineering-Economic Systems, Stanford University, August 1997.

[0262] [12] Schweizer, M. “A Guided Tour through Quadratic HedgingApproaches,” in Handbook in Mathematical Finance: Option Pricing,Interest Rates and Risk Management, E. Jouini, J. Civitanić, M. Musiela,eds. Cambrige University Press, Cambridge, 538-574, 1999.

[0263] [13] Luenberger, D. G. “Arbitrage and Universal Pricing,” toappear in the Journal of Economic Dynamics and Control.

[0264] [14] Heath, D., E. Platen, and M. Schweizer, “A Comparison of TwoQuadratic Approaches to Hedging in Incomplete Markets,” MathematicalFinance, Oct. 11, 2001, 385-413.

[0265] [15] Luenberger, D. G. “Pricing Derivatives of a Non-TradableAsset in Discrete Time”, In preparation.

[0266] Patents 5692233 Nov. 25, 1997 Garman 705/36 6173276 Jan 9, 2001Kant, et al 706/50

What is claimed is:
 1. A computer-implemented method for pricing afinancial derivative of a non-marketed variable x_(e), the methodcomprising: a) determining a market representative x_(m) useful indetermining a value of the financial derivative; b) retrievinginformation associated with the non-marketed variable x_(e) and themarket representative x_(m); c) calculating a solution to an equationinvolving a price of the financial derivative V(x_(e), t) defined as afunction of x_(e) and time t, wherein the equation comprises acoefficient involving the information associated with x_(e) and x_(m);and d) generating an output including the calculated price of thefinancial derivative.
 2. The method of claim 1 wherein the informationassociated with x_(e) and x_(m) comprises a drift rate of thenon-marketed variable x_(e), and a drift rate of the marketrepresentative x_(m).
 3. The method of claim 1 wherein the informationassociated with x_(e) and x_(m) comprises variances of the non-marketedvariable x_(e) and the market representative x_(m), and a covariancebetween the non-marketed variable x_(e) and the market representativex_(m).
 4. The method of claim 1 wherein the coefficient involving theinformation associated with x_(e) and x_(m) has the formμ_(e)−β_(em)(μ_(m)−r), where μ_(e) is a drift rate of the non-marketedvariable x_(e), μ_(m) is a drift rate of the market representativex_(m), r is an interest rate, and β_(em) is a factor derived from avariance of the market representative x_(m) and a covariance between thenon-marketed variable x_(e) and the market representative x_(m).
 5. Themethod of claim 1 wherein the equation is a modified Black-Scholesequation.
 6. The method of claim 5 wherein the modified Black-Scholesequation is obtained from a standard Black-Scholes equation byreplacing, in a term involving a first-order partial derivative ofV(x_(e), t) with respect to x_(e), a coefficient r, representing aninterest rate, by a coefficient involving the information associatedwith x_(e) and x_(m).
 7. The method of claim 1 wherein the equation is adiscrete-time equation involving V(x_(e), t) defined as a function ofx_(e) and discrete time points t=k.
 8. The method of claim 1 wherein themarket representative x_(m) comprises a marketed asset or combination ofsuch assets that is approximately most correlated with the non-marketedvariable x_(e).
 9. The method of claim 1 wherein the marketrepresentative x_(m) comprises a combination of multiple marketed assetsassociated with market sectors most closely associated with thenon-marketed variable x_(e).
 10. The method of claim 1 wherein themarket representative x_(m) comprises a marketed asset or combination ofsuch assets that is approximately equal to an overall market portfolio.11. The method of claim 1 further comprising calculating an optimalhedge.
 12. The method of claim 1 further comprising calculating aminimum variance of the error between an optimal hedge and thecalculated price of the financial derivative.
 13. The method of claim 1wherein the equation represents a risk-neutral discounted expected valueof cash flows of the financial derivative.
 14. The method of claim 13wherein a cash flow of the financial derivative is path-dependent. 15.The method of claim 1 applied to derivatives of a set of non-marketedvariables wherein the market representative x_(m) comprises acombination of multiple marketed assets, each most-correlated with adifferent non-marketed variable in the set of non-marketed variables.16. The method of claim 1 wherein the calculated price of the financialderivative includes cash flows at an intermediate time and a terminaltime.
 17. The method of claim 1 wherein drift rates, an interest rate,variances, and covariances of x_(e) and x_(m) either vary with time orare governed by stochastic processes.
 18. The method of claim 1 whereinthe cash flow depends on marketed variables as well as non-marketedvariables.
 19. The method of claim 1 wherein the equation involvesadditional non-marketed variables.
 20. The method of claim 1 wherein themarket representative is derived from a combination of multiple marketedvariables, and wherein x_(e) and the multiple marketed variables aregoverned by either geometric Brownian motion or alternative processes.21. A computer-implemented method of pricing a financial derivative of anon-marketed finite-state variable B, the method comprising: a)determining a market representative A associated with the non-marketedfinite-state variable B; b) calculating risk-neutral probabilities forthe non-marketed finite-state variable B using a binomial lattice modelassociated with the non-marketed finite-state variable B and the marketrepresentative A; c) calculating values of a price function V defined onthe lattice corresponding to the variable B; and d) generating from thecalculated values of the price function V an output including acalculated price of the financial derivative.
 22. The method of claim 21wherein the market representative A is determined to be approximatelyequal to at least one of a Markowitz portfolio, a market portfolio, anda market asset most correlated to the non-marketed finite-state variableB.
 23. The method of claim 21 further comprising calculating an optimalhedge.
 24. The method of claim 21 further comprising calculating aminimum variance of the error between an optimal hedge and thecalculated price of the financial derivative.
 25. The method of claim 21wherein a cash flow of the financial derivative is path-dependent. 26.The method of claim 21 wherein the binomial lattice model comprisestime-dependent lattice parameters of the variables A and B.